The Nonlocal Attraction-Repulsion Transport Equation With Power Kernels
MCML Authors
Massimo Fornasier
Prof. Dr.
Core PI
Lukang Sun
Dr.
Abstract
Massimo Fornasier
Prof. Dr.
Core PI
Lukang Sun
Dr.
Abstract
We study a nonlocal continuity equation on ℝd in which a probability density is driven by the competition between attraction toward a prescribed background measure ω and self-repulsion among particles, governed respectively by the power-law kernels ψa(x)=|x|1+a and ψr(x)=|x|1+r with exponents a,r∈[0,1). We establish global Lagrangian well-posedness via a squared-radius regularization, obtaining uniform L∞ and moment bounds, Wn,∞ regularity, and uniqueness in the Lagrangian class. When the initial data is compactly supported and attraction dominates (a>r, or a=r with ω(ℝd)>1), we prove that the support remains uniformly bounded at all time; a counterexample shows this fails for a=r>1. For the attractive-dominant nonquadratic range 0≤r≤a<1, we characterize zero-flux stationary states via a free-boundary problem involving a fractional Laplacian operator, reducing the stationarity condition to a fractional exterior Dirichlet problem. This characterization allow us to exhibit explicit examples of stationary measures in dimensions d∈{1,2,3}. Numerical particle simulations confirm agreement with the theoretical stationary profiles. Finally, we prove that every global solution with bounded energy and uniform moment bounds converges to a zero-flux stationary state.
BibTeXKey: FHS26